Design of a Socket and Spigot Cotter Joint for a Reversible Load of 20 kN

 

A cotter joint is a temporary joint used to connect two rods transmitting axial motion without rotation. It consists of three main parts: the socket, spigot, and cotter. The cotter is designed to be the weakest part for easy replacement. Below is the step-by-step design procedure for a cotter joint to carry a reversible load of 20 kN.

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Step 1: Selection of Material

• Material: C20 steel (mild steel)

• Properties:

  • Ultimate tensile strength ($S_{ut}$) = 480 MPa

  • Yield strength ($S_{yt}$) = 260 MPa

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Step 2: Selection of Factor of Safety

• Factor of Safety (n): 4 (for reversible load)

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Step 3: Determination of Allowable Stresses

1. Design Tensile Strength ($S_{dt}$):

   $$S_{dt}=\frac{S_{yt}}{n}=\frac{260}{4}=65\text{MPa}$$

2. Design Shear Strength ($S_{ds}$):

   $$S_{ds}=0.5\times S_{dt}=0.5\times 65=32.5\text{MPa}$$

3. Design Crushing Strength ($S_{dcr}$):

   $$S_{dcr}=1.3\times S_{dt}=1.3\times 65=84.5\text{MPa}$$

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Step 4: Design of Spigot End

4a: Diameter of Rod ($d$)

The rod is subjected to tensile stress. The diameter ($d$) is calculated as:

$${\sigma }_t=\frac{W}{\frac{\pi }{4}{d}^2}$$$$65=\frac{20,000}{\frac{\pi }{4}{d}^2}$$$$d^2=\frac{20,000\times 4}{\pi \times 65}=392.7$$$$d=\sqrt{392.7}=19.8\text{mm}\approx 20\text{mm}$$

4b: Outside Diameter of Spigot ($d_1$)

The spigot is subjected to tensile stress across the slot. The diameter ($d_1$) is calculated as:

$${\sigma }_t=\frac{W}{\frac{\pi }{4}(d_1^2-d^2)}$$$$65=\frac{20,000}{\frac{\pi }{4}(d_1^2-20^2)}$$$$d_1^2-400=\frac{20,000\times 4}{\pi \times 65}=392.7$$$$d_1^2=792.7⟹d_1=28.2\text{mm}\approx 30\text{mm}$$

4c: Thickness of Cotter ($t$)

The cotter is subjected to crushing stress. The thickness ($t$) is calculated as:

$${\sigma }_{cr}=\frac{W}{t\cdot d_1}$$$$84.5=\frac{20,000}{t\cdot 30}$$$$t=\frac{20,000}{84.5\times 30}=7.88\text{mm}\approx 8\text{mm}$$

4d: Distance from End of Slot to End of Spigot ($a$)

The spigot is subjected to double shear. The distance ($a$) is calculated as:

$$\tau =\frac{W}{2\cdot a\cdot t}$$$$32.5=\frac{20,000}{2\cdot a\cdot 8}$$$$a=\frac{20,000}{2\times 32.5\times 8}=38.46\text{mm}\approx 40\text{mm}$$

4e: Diameter of Spigot Collar ($d_2$) and Thickness ($t_1$)

The spigot collar is subjected to shear stress. The diameter ($d_2$) and thickness ($t_1$) are calculated as:

$$\tau =\frac{W}{\pi \cdot d_2\cdot t_1}$$

Assume $t_1=10\text{mm}$:

$$32.5=\frac{20,000}{\pi \cdot d_2\cdot 10}$$$$d_2=\frac{20,000}{\pi \times 32.5\times 10}=19.6\text{mm}\approx 20\text{mm}$$

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Step 5: Design of Socket

5a: Outside Diameter of Socket ($d_3$)

The socket is subjected to tensile stress across the slot. The diameter ($d_3$) is calculated as:

$${\sigma }_t=\frac{W}{\frac{\pi }{4}(d_3^2-d_1^2)}$$$$65=\frac{20,000}{\frac{\pi }{4}(d_3^2-30^2)}$$$$d_3^2-900=\frac{20,000\times 4}{\pi \times 65}=392.7$$$$d_3^2=1292.7⟹d_3=35.9\text{mm}\approx 40\text{mm}$$

5b: Diameter of Socket Collar ($d_4$)

The socket collar is subjected to crushing stress. The diameter ($d_4$) is calculated as:

$${\sigma }_{cr}=\frac{W}{(d_4-d_1)\cdot t}$$$$84.5=\frac{20,000}{(d_4-30)\cdot 8}$$$$d_4-30=\frac{20,000}{84.5\times 8}=29.6\text{mm}$$$$d_4=59.6\text{mm}\approx 60\text{mm}$$

5c: Axial Distance from Slot to End of Socket Collar ($c$)

The socket collar is subjected to double shear. The distance ($c$) is calculated as:

$$\tau =\frac{W}{2\cdot c\cdot t}$$$$32.5=\frac{20,000}{2\cdot c\cdot 8}$$$$c=\frac{20,000}{2\times 32.5\times 8}=38.46\text{mm}\approx 40\text{mm}$$

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Step 6: Design of Cotter

6a: Width of Cotter ($b$)

The cotter is subjected to bending stress. The width ($b$) is calculated as:

$${\sigma }_b=\frac{3W\cdot l}{2\cdot b\cdot t^2}$$

Assume $l=50\text{mm}$:

$$65=\frac{3\times 20,000\times 50}{2\cdot b\cdot 8^2}$$$$b=\frac{3\times 20,000\times 50}{2\times 65\times 64}=36.1\text{mm}\approx 40\text{mm}$$

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Final Design Specifications:

1. Spigot:

   • Diameter of rod ($d$) = 20 mm

   • Outside diameter of spigot ($d_1$) = 30 mm

   • Diameter of spigot collar ($d_2$) = 20 mm

   • Thickness of spigot collar ($t_1$) = 10 mm

   • Distance from slot to end of spigot ($a$) = 40 mm

2. Socket:

   • Outside diameter of socket ($d_3$) = 40 mm

   • Diameter of socket collar ($d_4$) = 60 mm

   • Axial distance from slot to end of socket collar ($c$) = 40 mm

3. Cotter:

   • Thickness ($t$) = 8 mm

   • Width ($b$) = 40 mm

   • Length ($l$) = 50 mm

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This design ensures the cotter joint can safely carry a reversible load of 20 kN.